Question

use the triangle proportionality theorem to solve for x. given that hi=31, ji=6, and gf=12 round your answer to the nearest whole number.

Explanation:

Step1: Apply the triangle proportionality theorem

According to the triangle proportionality theorem, if a line is parallel to one side of a triangle and intersects the other two sides, then it divides those sides proportionally. So, \(\frac{JI}{JF}=\frac{HI}{HG}\). Let \(HG = x\). Since \(JF=JI + IF\) and \(IF = JI\) (because of the mid - point - like division shown by the red dots), \(JF = 2JI\). Then \(\frac{JI}{2JI}=\frac{HI}{HI + x}\). Simplifying \(\frac{1}{2}=\frac{31}{31 + x}\).

Step2: Cross - multiply

Cross - multiplying gives \(31 + x=2\times31\).

Step3: Solve for \(x\)

\(31+x = 62\), then \(x=62 - 31=31\).

Answer:

\(31\)